Sheffer stroke

In the late 19th century and early 20th century, Charles Sanders Peirce and H.M. Sheffer independently discovered that a single binary logical connective suffices to define all logical connectives (they are each functionally complete). Two such connectives are

• $\uparrow$: the Sheffer stroke (sometimes denoted by $|$) and

• $\downarrow$: the Peirce arrow (sometimes denoted by $\perp$).

The Sheffer stroke is defined by the truth table

$P$	$Q$	$P\uparrow Q$
F	F	T
F	T	T
T	F	T
T	T	F

$Observethat$P↑Q$istrueifandonlyifeither$P$or$QP↓Q$istrueifandonlyifboth$P$and$Q$arefalse.Forthisreason,thePeircearrowissometimescalled\mathit{\text{<em class="ltx\_emph ltx\_font\_italic">joint denial</em>}}or\text{<em class="ltx\_emph ltx\_font\_italic">NOR</em>}.ToshowthesufficiencyoftheShefferstroke,allwehavetodoisdefineboth$¬$and$∨$intermsof$↑$.Theproposition$P↑P$assertsthateither$P$isfalse,or$P$isfalse;thuswecandefine$¬$by$¬P := P↑P$.Wedefine$∨$by\text{ P ∨ Q := ( P ↑ P ) ↑ ( Q ↑ Q ) , }sincethisassertsthateither$P↑P$isfalse(thatis,that$P$istrue)orthat$Q↑Q$isfalse(thatis,that$Q$istrue).WecanshowthesufficiencyofthePeircearrowinasimilarway.Define\text{ ⁢ ¬ P := P ↓ P }and\text{ P ∨ Q := ( P ↓ Q ) ↓ ( P ↓ Q ) . }Thisexpressionassertsthat$P↓Q$isfalse,thatis,thatitisfalsethatboth$P$and$Q$arefalse.ByDeMorga{n}^{\prime }slaw,thisisequivalenttoassertingthatatleastoneof$P$and$Q